Detecting Earth-Like Exoplanets Using High-Dispersion Nulling Interferometry

Master Thesis

Detecting Earth-like exoplanets using high-dispersion nulling interferometry

Supervisors : Author : Jean-Philippe BERGER Germain Garreau Guillermo MARTIN Xavier BONFILS

Conﬁdentiality : no Year 2020/2021 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Acknowledgment : I truly believe that in research and for many other fields, no results can be allocated to only one person but are instead the outcome of a team work, a collaboration of individuals and a lot of discussions. That is why I would like to take this opportunity to acknowledge all the persons who participated in this project and spent time helping me or sharing their knowledge with me. Firstly I want to thank everyone from the IPAG for their hospitality even during these times of pandemic implying a lower activity. I want to thank the PhD/master students who worked with me and shared tea breaks with me. I specifically thank Guillaume for using his precious time to help me resolve some of my problems when my supervisors were not available. Moreover I want to thank my supervisors Guillermo and Xavier for all the support they offered me to make sure that I could always progress in my project. And finally I want to thank my supervisor Jean-Philippe for his unlimited support and all the time he spent helping me despite his own health. This master thesis was a wonderful opportunity to work in a field I never expected to reach as an engineering student and that I wish to pursue as far as possible.

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The Grenoble Institute of Planetology and Astrophysics (IPAG) is the result of the merging between the Grenoble Laboratory of Astrophysics (LAOG) and the Grenoble Laboratory of Planetology (LAP) in 2011. It is part of the French National Centre for Scientiﬁc Research (CNRS) and the Grenoble-Alpes University. The institute gather an average of 170 workers: researchers, engineers, PhD students, etc...

The laboratory is famous for its contribution in many instruments on the ground such as GRAV- ITY, PIONIER or SPHERE.

The institute is divided into 7 teams of research studying various ﬁelds such as stellar formation, plasmas or astrochemistry. My master thesis was part of the CHARM (Contrast High Angular Resolution spectro-iMaging) and EXOPLANETES teams.

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Contents

Glossary v

List of Figures vi

1 Introduction 1 1.1 Context ...... 1 1.2 Nulling interferometry: Bracewell’s principle ...... 1 1.3 The high-dispersion spectroscopy ...... 3 1.4 Goals of this study ...... 4

2 Simulation of the detected signal 5 2.1 Objectives ...... 5 2.2 The emitted signal ...... 5 2.3 The shot noise ...... 6 2.4 Implementation of a noise ...... 6 2.5 The background signal ...... 6 2.5.1 Definition ...... 6 2.5.2 The thermal emission ...... 7 2.5.3 The local zodiacal emission ...... 8 2.5.4 The Exozodiacal Emission ...... 9 2.5.5 Support for the results ...... 9 2.6 The interferometer limitations ...... 9 2.6.1 Limitations for the star light attenuation ...... 9 2.6.2 The stellar leakage ...... 10 2.7 The detector noise ...... 10 2.7.1 Definition ...... 10 2.7.2 The readout noise ...... 11 2.7.3 The dark current ...... 11 2.7.4 Example of detector noise ...... 11 2.7.5 Quantum efficiency ...... 11

3 Extracting the exoplanet signal 12 3.1 The cross-correlation function ...... 12 3.2 The Doppler shift of the spectra ...... 12 3.3 Correction of the correlation peak ...... 13 3.4 The high-pass ﬁlter ...... 15 3.5 Signal-to-noise ratio of the correlation peak ...... 15

4 Performance simulation 16 4.1 Design of the instrument ...... 16 4.2 Conditions of the simulation ...... 17 4.2.1 Object ...... 17 4.2.2 Instrument ...... 17 4.2.3 Software ...... 19 4.3 Results of the simulation ...... 20

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4.3.1 Without stellar leakage ...... 20 4.3.2 Comments ...... 23 4.3.3 With stellar leakage ...... 23 4.4 Conclusion ...... 24

5 Study of a photonic device for the nulling 25 5.1 Introduction ...... 25 5.2 Component description ...... 25 5.3 Experimental setup ...... 26 5.4 Interferogram in the monochromatic case ...... 27 5.5 The contrast ...... 28 5.5.1 The chromatism ...... 29 5.5.2 The photometric equilibrium ...... 31 5.5.3 The relative phase instability ...... 31 5.6 Comments ...... 32

6 Conclusion 33

Appendices 36

Abstract 39

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Glossary

• Astronomical Unit (A.U) : 1A.U ' 1,5.1011m • Electromagnetic wavelength: λ • ELT: Extremely Large Telescope • ESA: European Space Agency • ESO: European Southern Observatory • Jansky (Jy) : 1Jy = 10−26 W/m2/Hz • JWST: James Webb Space Telescope • LIFE: Large Interferometer For Exoplanets • Light speed: c=3.108m/s • NASA: National Aeronautics and Space Administration • NIR/MIR: Near-Infrared/Mid-Infrared • Parsec (pc) : 1pc ' 3.1016m • SNR: Signal-to-Noise Ratio • Spectral bands: – J-band: λ ∈[1.143-1.375]µm – H-band: λ ∈[1.413-1.808]µm – K-band: λ ∈[1.996-2.382]µm – L-band: λ ∈[3.42-4.12]µm – M-band: λ ∈[4.6-5.0]µm – N-band: λ ∈[7.5-14.5]µm

λ • Spectral resolution: R = ∆λ • SPHERE: Spectro-Polarimetric High-contrast Exoplanet REsearch

8 • Sun radius: R ' 7.10 m • TPF-I: Terrestrial Planet Finder-Interferometer • VLT: Very Large Telescope

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List of Figures

1 Coronagraphic image of AB Pictoris showing a companion (bottom left). The data was obtained using a 1.4 arcsec occulting mask on top of AB Pictoris. Source: ESO...... 1 2 A schematic cartoon showing the key functional elements of a simple 2-element interferometer. Light from the two collectors travels along the optical paths d1 and d2 and is interfered and detected at the “beam combiner” in the centre of the figure. Source: [1]...... 2 3 Schematic example of an interferometer output without π-phase shift for a star and an off-centered exoplanet...... 3 4 Same as Fig.3 but with a π-phase shift in one of the interferometer arms...... 3 5 Example of light diffraction with a grating. The diffracted signal is sent to the detector in order to obtain the photon spectral distribution of the signal. . . . .3 6 Example of differences between the star spectrum and the addition of the star and exoplanet spectrum...... 4 7 Schema of the simulated situation without detailing the software part...... 5 8 Emitted spectra of the star and the exoplanet with a spectral resolution of R=103 from the PSG...... 5 9 Four examples of configurations from nano- to medium-size satellites. Source: [2].7 10 Example for a setup of two telescopes with diameter 0.5m, a resolution of R=103 and an optical train temperature of Toptical=60, 100 and 150K ...... 7 11 Example of JWST Backgrounds Tool output. Source: JWST Backgrounds Tool.8 12 Zodiacal emission spectrum according to the results of the JWST Backgrounds Tool...... 8 13 Simulation for an exoplanet orbiting Proxima Centauri with a set of two telescopes with diameter D=1m and Toptical=150K. Source: [2]...... 9 14 Our simulation for an exoplanet orbiting Proxima Centauri with two telescopes of D=0.5m and Toptical=150K...... 9 15 Schema of the stellar leakage phenomenon for an interferometer. Source: [3]. . . 10 16 Example of the instrumental noise impact on the exoplanet spectrum...... 11 17 Schema of the detailed software we realized with its different parts...... 12 18 Example of Doppler shift depending on the radial velocity of the host star. In our case we consider the difference of velocity between the exoplanet and the host star. Source: ESO...... 13 19 Correlation peak with an exoplanet (blue) and without (orange). The case without exoplanet is a false positive...... 14 20 Same as in Fig.19 but with a velocity difference ∆v=50km/s between the star and the exoplanet...... 14 21 Normalized correlations in the case of Fig.20 (orange) and with the correction from (11) (blue)...... 14 22 Example of correlation function without high-pass filtering. The large spectral structures are overwhelming...... 15 −1 23 Example of correlation function with high-pass filtering with νc=50µm . The correlation peak appears...... 15

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24 Schema of the SNR calculation protocol...... 16 25 Four examples of configurations from nano- to medium-size satellites. Source: [2]. 18 26 NIRSpec NRS1 and NRS2 detector performance summaries. Source: NIRSpec Detector Performance...... 18 27 Evolution of the SNR with the cutoff resolution (1/fc) for spectral resolution R=103 (top,left), 104 (top,right), 105 (bottom,left) and 106 (bottom,right). The study is made in the K-band with t=24hrs, 1/ρ=10−6 and the shot noise as the only source of noise...... 19 28 Example of a full correlation function...... 20 29 Evolution of the SNR with the spectral resolution for three sources of photoelectronic noise. We consider the star light attenuation C=1/ρ=10−6 (no stellar leakage)...... 21 30 Evolution of the quadratic difference between the ideal case and with one source of noise. The ideal case is added as a reference but has no meaning here. . . . . 21 31 Evolution of the SNR with the spectral resolution for different background sources. We consider the star light attenuation C=1/ρ=10−6 (no stellar leakage). . . . . 21 32 Evolution of the quadratic difference between the ideal case and with one source of noise. The ideal case is added as a reference but has no meaning here. . . . . 21 33 Evolution of the SNR in (C,R) without considering any source of noise...... 22 34 Evolution of the SNR in (C,R) considering only the shot noise. The stellar leakage is not considered...... 22 35 Evolution of the SNR in (C,R) considering only the dark current. The result is independent from C...... 22 36 Evolution of the SNR is (C,R) when considering all the noises except the stellar leakage...... 23 37 Evolution of the SNR in (C,R) considering only the shot noise...... 24 38 Evolution of the SNR in (C,R) considering the shot noise and the stellar leakage. 24 39 Schema of the 2T ABCD photonic device with two sets of electrodes. Source: [4] 25 40 Schema of the phase difference between the four outputs of the device. Source: [5]. 26 41 Schema of the SILVI experimental setup. Source: [6] ...... 26 42 ASE spectrum from the constructor...... 27 43 Interferograms for the four outputs in the Laser case with external modulation. 28 44 Corrected interferograms for outputs in Fig.43 using (19)...... 29 45 Example for a polychromatic source. It shows the individual monochromatic re- sponses (left) and the resulting fringe pattern for a polychromatic source (right). Source:[1]...... 30 46 Observed interferogram for the SLED source. We can observe the impact of the coherency envelope...... 30 47 Evolution of the null depth with the intensity mismatch between the two inputs. 31 48 Evolution of the detected signal from the four outputs of the component over time without any modulation...... 32 49 Example for a setup of two telescopes with diameter 1m, a resolution of R=103, an integration time of t=100hrs and an optical train temperature of Toptical=60, 100 and 150K ...... 36 50 Evolution of the SNR with the spectral resolution considering the shot noise with and without the quantum efficiency. The star light attenuation is at C=1/ρ=10−6. 37

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51 Evolution of the quadratic diﬀerence between the ideal case and with the shot noise (blue) and between the shot noise without and with quantum eﬃciency (red). The ideal case is added as a reference but has no meaning ...... 37

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1 Introduction

1.1 Context The Large Interferometer For Exoplanets (LIFE) is a proposition for a space mission designed to characterize terrestrial exoplanets atmospheres, assess their habitability and the presence of life. It follows the two previous space mission proposals of ESA/Darwin and NASA/TPF-I in 1993 and 2002 but cancelled in 2007 and 2011 respectively. Like the Darwin study before, LIFE currently foresees a nulling interferometry concept for the detection of exoplanets. Indeed, one of the main problems in exoplanets detection is the very high contrast between the star and the object signal (star/planet∼ 106-107 for terrestrial exoplanets in the MIR [7]). This level of contrast prevents any observation with the current technologies of detection, therefore an efficient star light attenuation is necessary to observe the signal from possible star companions. Reducing the star-planet contrast is a difficult task with two major solutions used in the field of astronomical instrumentation. The first and oldest method is the coronagraphy which consists of blocking the light from the star with a physical obstacle, the coronagraph.

Figure 1: Coronagraphic image of AB Pictoris showing a companion (bottom left). The data was obtained using a 1.4 arcsec occulting mask on top of AB Pictoris. Source: ESO.

This method, demonstrated in 1931 by the astronomer B.Lyot, is widely used in astronomy for the detection of widely separated massive planets in the optical or NIR spectral band (with the SPHERE instrument at the VLT for example). However, the detection of Earth-like planets is hampered by the combination of the very low planet to star ﬂux ratio and the close angular separation that prevents them to be detected with standard telescopes. In order to reach a higher angular resolution and detect them, the new method of nulling interferometry has been designed by R.N.Bracewell in 1978.

1.2 Nulling interferometry: Bracewell’s principle With the current technologies, the angular resolution R with a single telescope is limited. Only the wide separated exoplanets can be detected even when considering the new giant telescopes

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with a very important diameter D (the ELT with D∼40m, estimated to cost 1.15 B$ in 2017).

λ R ∼ (1) D The detection of inner exoplanets and especially in the habitable zone requires the search for higher angular resolution. Because increasing the diameter of a single telescope would be too expensive and technically problematic, we need to overcome it by using several telescopes and the interferometry method. The interferometry consists of superimposing electromagnetic waves in order to make them interfere. Every system able to realize these interferences is called an interferometer, the ﬁrst ones being the Fizeau interferometer (1851) and the Michelson interferometer (1881).

Figure 2: A schematic cartoon showing the key functional elements of a simple 2-element interferometer. Light from the two collectors travels along the optical paths d1 and d2 and is interfered and detected at the “beam combiner” in the centre of the ﬁgure. Source: [1].

In the example Fig.2, the separation between the two telescopes B is the Base (or Baseline). The angular resolution of such setup is equivalent to one telescope of diameter B: λ R ∼ (2) B This allows to improve the angular resolution by increasing the telescopes separation. New instruments such as GRAVITY at ESO are using interferometry to reach high levels of

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angular resolution to observe the environment around the center of the Milky Way for example.

The nulling interferometry is a speciﬁc conﬁguration where we add an achromatic π-phase shift after one of the telescopes to generate a destructive interference.

Figure 3: Schematic example of an in- Figure 4: Same as Fig.3 but with a terferometer output without π-phase π-phase shift in one of the interferom- shift for a star and an oﬀ-centered ex- eter arms. oplanet.

The relative path difference in Fig.3-4 corresponds to the difference of optical path for the light between the two arms of the interferometer in Fig.2. When centered on the star, its light is interfering destructively which causes the star attenuation in Fig.4. Because the exoplanet is off-centered, its light doesn’t interfere destructively. Therefore the exoplanet is not attenuated which means it can now be detected for a sufficient stellar attenuation.

1.3 The high-dispersion spectroscopy After the light is transmitted through the interferometer, we need to obtain its spectrum. For this, we can use a spectrometer (for example a grating) to diﬀract the light. This way we are able to detect the spectral density of photons:

Figure 5: Example of light diﬀraction with a grating. The diﬀracted signal is sent to the detector in order to obtain the photon spectral distribution of the signal.

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High-dispersion spectroscopy consists of obtaining spectra with high spectral resolution R: λ R ∼ (3) ∆λ The interest of high spectral resolution is to reveal thin absorption lines from the exoplanet spectrum in the detected signal (see Fig.6).

Figure 6: Example of diﬀerences between the star spectrum and the addition of the star and exoplanet spectrum.

The detected spectrum corresponds to the top image in Fig.6 where absorption lines from the planet are spread among the star spectrum. Therefore the idea is to combine them so we can disentangle and isolate the planet signal and increase its overall detectability.

1.4 Goals of this study Several papers like [8] are explaining the benefits of combining coronagraphy and high spectral dispersion. Such method would be able to relax the star light attenuation required to detect Earth-like exoplanets, especially for ground-based instruments such as the ELT. The first goal of this project is then to evaluate quantitatively the interest of coupling nulling interferometry and high-dispersion spectroscopy for the detection of low-mass rocky planets (Sections 2-4). This work will be inspired by the precursor space mission proposal in [2]. For the last two decades IPAG has been exploring the applications of photonic technologies for precision interferometry. This effort leads to laboratory prototyping and design of precursor instruments. The current research focuses on exploring the potential and limitations of photonics based on LiNBO3 crystals. For that purpose we have tested the first two telescope interferometric combiner with internal active modulation described in [4]. The second goal of the project thus consists of characterizing this component on an optical bench and study its limitations (Section 5).

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2 Simulation of the detected signal

2.1 Objectives Let us consider a schema of the situation we want to simulate:

Figure 7: Schema of the simulated situation without detailing the software part.

The object we observe is a planetary system with a star and an exoplanet orbiting it. The instrument is made of two telescopes to collect photons from the object, a beam combiner to realize the nulling interferometry, a spectrometer for the high spectral dispersion and a photo-detector to convert photons into electrons. The software part aims at extracting the planet signal and its intensity and returns its Signal- to-Noise Ratio (SNR). The details for this procedure will be studied later. For now, the goal is to simulate each part of the situation in Fig.7.

2.2 The emitted signal The spectra for the star and the exoplanet are complex to simulate because they need to take into account a lot of phenomena such as the thermal emission, the star light reﬂection, the molecular absorption, etc...Because we want to make a realistic simulation, we will use the Planetary Spectrum Generator (PSG) from [9] to obtain the ideal emitted spectra of both the star and the exoplanet in orbit around it at any spectral resolution R. A previous work by Mathilde Gardies already used the PSG software to generate the spectra for a star similar to Proxima Centauri at 1.29pc and an Earth-like exoplanet orbiting at 0.032 A.U from it. This system is very close to the one considered in [2]. Fig.8 gives an example of the generated spectra by the PSG for a certain spectral resolution.

Figure 8: Emitted spectra of the star and the exoplanet with a spectral resolution of R=103 from the PSG.

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2.3 The shot noise One source of noise that is irreducible is the shot noise from the photon distribution. When we want to count the precise number of detected photons, the value we obtain always comes with an uncertainty following a Poisson statistic:

λke−λ f(k; λ) = (4) k! with λ the real number of photons and k the number of detected photons. The variance of such distribution is equal to the real√ number of photons λ, which means that the standard deviation from this value is equal to λ. As a consequence, a√ strong photonic signal will have a better Signal-to-Noise Ratio (SNR) because SNR= √λ = λ. The only way to reduce the impact of the shot noise then is to have λ a stronger signal and collect more photons from the planet.

2.4 Implementation of a noise For every sources of noise X that we catalog below, they will generate additional photons/photo- electrons with a Poisson statistic. In each case, the number of additional photons/photo- electrons NX can be expressed with a constant mean value < NX > and a random number ∆NX : NX =< NX > +∆NX (5)

A property of a Poisson distribution is that its variance VX has the same value than its expected value EX which also corresponds to < NX >. Knowing either VX or EX from the parameters of the simulation then allows us to correct < NX > in the detected signal. Only the random value ∆NX will thus remain and hamper the planet signal. To model it, we can generate a Poisson distribution NX with a variance/expected value equals to < NX >, ∆NX then corresponds to:

∆NX = NX − < NX > (6) which is eventually added to the detected photonic signal to simulate the impact of the noise X.

2.5 The background signal 2.5.1 Deﬁnition When observing the planetary system, in addition to the signal from the star and the planet, we will also detect photons from other objects. The term "Background" amalgamates all the sources for these photons. Many studies in exoplanets detection such as [2] are considering three main sources contributing to the background: • The thermal emission of the optical train • The local zodiacal disk • The exozodiacal disk

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The zodiacal disk is a disk of grains and dust (∼1-100µm diameter) orbiting around the Sun. The exozodiacal disk corresponds to its equivalent in the observed planetary system. These signals will generate an additional shot noise when detected. By evaluating their intensities, we will be able to quantify the noise level from the background.

2.5.2 The thermal emission The optical train represents all the components of our instrument (lenses,mirrors,...) which are used to guide the incoming light. The thermal emission corresponds to photons coming from the optical train of the instrument emitting like a black body with a certain temperature. In the article [2], the authors consider three diﬀerent temperatures.

Figure 9: Four examples of conﬁgurations from nano- to medium-size satellites. Source: [2].

According to Fig.9, the temperature can go from 150K to 60K in the best scenario. For these temperatures, we compute the number of detected photons using the Planck’s law of blackbody radiation. This process is further described in Appendix.1.

Figure 10: Example for a setup of two telescopes with diameter 0.5m, a resolution of R=103 and an optical train temperature of Toptical=60, 100 and 150K

We see that going in the MIR is interesting in terms of planet/star contrast but it makes the cooling temperature critical. Indeed the photon ﬂux can even overcome the ﬂux from the

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star at some wavelength, preventing any possible observation. The choice of temperature in Fig.10 corresponds to the three examples of space missions from Fig.9 with CubeSat (150K), PROBA-size (100K) and FKSI-concept (60K).

2.5.3 The local zodiacal emission The zodiacal emission corresponds to the light coming from the disk orbiting the Sun. The grains in the disk behave like black bodies, emitting light at a temperature around 300K in the MIR, as well as scattering light from the Sun. The zodiacal and exozodiacal disks are generally radiating in the habitable zone of their system and farther, making the observation of exoplanets in this region more diﬃcult when the disk is brighter than the planet itself. The simulation for the spectrum of this source is complex but it has already been made for the JWST mission following the COBE model in [10]. The program called "JWST Backgrounds Tool" is free and allows modeling of the major background sources spectra.

Figure 11: Example of JWST Backgrounds Tool output. Source: JWST Backgrounds Tool.

We will use the output ’Zodi’ in Fig.11 to estimate the impact of the zodiacal emission in our simulation. By converting the units into a number a photons, we can compare the results with the other spectra from the star and planet.

Figure 12: Zodiacal emission spectrum according to the results of the JWST Backgrounds Tool.

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2.5.4 The Exozodiacal Emission The intensity of an exozodiacal disk can vary up to 100 times the local one but most of them are limited to 10 times. Its emission is very important in the case of a direct observation with a coronagraph because its presence can quickly hamper the planet signal. However for an intensity of less than 20 times the local one, the impact of the exozodi is negligible for an interferometer case [11]. We are therefore not studying its impact.

2.5.5 Support for the results Our results for the star, the exoplanet, the thermal emission and the zodiacal light have already been simulated in [2] with similar parameters but a diﬀerent software.

Figure 13: Simulation for an exo- Figure 14: Our simulation for an planet orbiting Proxima Centauri with exoplanet orbiting Proxima Centauri a set of two telescopes with diameter with two telescopes of D=0.5m and D=1m and Toptical=150K. Source: [2]. Toptical=150K.

If we make a qualitative comparison between the two results in Fig.13-14, it supports the fact that for wavelengths higher than 7µm, the thermal emission becomes dominant for 150K compared to the exoplanet signal and equivalent to the star signal. The results in Fig.13 also conﬁrm that the exozodiacal emission for an interferometer setup is negligible compared to the other backgrounds. Finally, the zodiacal emission has a low impact in NIR compared to the exoplanet but increases in intensity as we move towards the MIR. A study of its impact is thus necessary in our project.

2.6 The interferometer limitations 2.6.1 Limitations for the star light attenuation The eﬀective star light attenuation C that an interferometer can achieve depends on the equation from [12]:  2 !2  1 2 π θdia C = σφ + + 1/ρ (7) 4 4 λsh/b

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with σφ the variance in optical path delay, θdia the angular diameter of the star, λsh the shortest wavelength of the bandpass, b the baseline of the interferometer and ρ the rejection ratio. This equation shows that the eﬀective star light attenuation C is depending on three terms, the highest one.s will cap the maximum attenuation achievable:

The ﬁrst term will be ignored (σφ=0) since we are considering that the optical path delay is always perfectly stabilized on the center of the star thanks to a phase control. The second term represents the stellar leakage from the observed star and the interferometer. It depends on the observed star, the baseline of the interferometer and the number of telescopes. The third term 1/ρ corresponds to the null depth of the beam combiner. This will be an important parameter in our simulations which we will use and study in the following chapters.

2.6.2 The stellar leakage Because a star is a resolved source, meaning that its angular diameter is not negligible, we have to consider the part of the star that is not centered on the destructive interference.

Figure 15: Schema of the stellar leakage phenomenon for an interferometer. Source: [3].

In Fig.15, we see that the edges of the star are not fully attenuated by the interference and a part of the star light can still be transmitted. This is the phenomenon called stellar leakage and it caps the star light attenuation in (7). This eﬀect can be limited by increasing the number of telescopes we are using in our interferometer. According to [13], this makes the destructive interference wider and therefore increases the angular null depth.

2.7 The detector noise 2.7.1 Deﬁnition The detector noise corresponds to the electrons that are generated by the detector. This signal holds no spectral information so we need to consider its variance in order to quantify its noise

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level. We will consider two major sources of detector noise that are typically encountered with detectors: • The readout noise • The dark current

2.7.2 The readout noise When reading the value of a detector, a random number of electrons can be added to the electronic signal because of the error in the ampliﬁcation and digitization of the electronic charge. This ﬂuctuation is called readout noise and its variance is a constant known from the constructor and expressed in number of electrons.

2.7.3 The dark current Even in the absence of incoming photons, the charge of the detector will still increase because of the thermal ﬂuctuations in the material that can emit photons passively. This phenomenon called dark current has a known variance expressed in electrons/s.

2.7.4 Example of detector noise

− − If we take for variances Vread=100 e and Vdark=1e /s × t with t, the integration time. Then for one hour of exposure, we have an example of the instrumental noise impact.

Figure 16: Example of the instrumental noise impact on the exoplanet spectrum.

2.7.5 Quantum eﬃciency This phenomenon is diﬀerent from the others since it will not generate additional electrons but rather reduce the detected signal intensity. The purpose of the detector is to convert the photons into electrons (or photo-electrons). The photons are increasing the charge of the detector by exciting hole-electron pairs in the semi- conductor. This charge is then released when reading the value of the detector. However this

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process is not perfect and part of the photons are not converted into photo-electrons because of phenomena like light reﬂection or recombination. This is called the quantum eﬃciency and corresponds to a parameter ηeff between 0 and 1.

3 Extracting the exoplanet signal

3.1 The cross-correlation function When the signal is detected and transmitted via photo-electrons into the software (Fig.7), the objective is to treat the electronic signal and extract the planet Signal-to-Noise Ratio (SNR). To do this, we developed a software that realizes a cross-correlation function based on the article [8]. The basic idea is to make a cross-correlation between the detected spectrum and a template spectrum of an exoplanet. If parts of the detected spectrum shares information with the template, a correlation peak will emerge with intensity proportional to the planet signal. In order to use this method, we will follow these main steps: 1. Consider a detected spectrum with high spectral resolution R, 2. Correct the detected spectrum with the non-attenuated star spectrum, 3. Synthesize a template spectrum of an Earth-like exoplanet, 4. Filter the continuum part and low-frequency structures of both spectra, 5. Make a cross-correlation between both spectra and plot the results, 6. Calculate the signal-to-noise ratio of the correlation peak. We can then show an explicit description of the software from Fig.7 which summarizes these steps.

Figure 17: Schema of the detailed software we realized with its diﬀerent parts.

3.2 The Doppler shift of the spectra In order to realize the cross-correlation, we need to shift one of the spectra from the other one and then compute the correlation. One shift value then corresponds to one point in the correlation function. We also have to anticipate a diﬀerence in radial velocity ∆v between the planet and the star

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that can generate an oﬀset corresponding to a Doppler shift between the spectra. This would shift the wavelength λ of the spectrum by a multiplication factor α from the original wavelength λ0: ∆v ! λ = α.λ = 1 + .λ (8) shift 0 c 0

Figure 18: Example of Doppler shift depending on the radial velocity of the host star. In our case we consider the diﬀerence of velocity between the exoplanet and the host star. Source: ESO.

As a consequence, a positive velocity diﬀerence (∆v > 0) means that the exoplanet is moving away from us faster than the host star. Therefore we have a red shift of the exoplanet spectrum (see Fig.18). In the opposite case, for ∆v < 0 we have a blue shift of the exoplanet spectrum. For any points i in our two spectra, the i+1 point needs to respect the following condition:

λi+1 = α.λi (9) with α a constant common for both the star and exoplanet spectra. This way, shifting a spectrum from one another by n points is equivalent to applying a factor αn to the wavelengths. This is also equivalent to a velocity diﬀerence ∆v of:

∆v = c.(αn − 1) (10)

In the following correlation functions, the shifting parameter of the spectrum in abscissa will always be given in terms of velocity diﬀerence.

3.3 Correction of the correlation peak It is possible when considering certain spectral bands to see a correlation peak between the template spectrum and the star spectrum only. Because this peak is not caused by the presence of a planet, this correlation is a false positive. This false positive may be a problem if we want to estimate the exoplanet SNR.

13 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Figure 19: Correlation peak with Figure 20: Same as in Fig.19 but with an exoplanet (blue) and without (or- a velocity diﬀerence ∆v=50km/s be- ange). The case without exoplanet is tween the star and the exoplanet. a false positive.

The origin for this false positive is still uncertain but it is likely to come from the common molecular absorption lines that are due to the planet reflection of the host star light. This reflection is simulated in the template spectrum of the exoplanet and therefore it could give rise to this correlation. Since we have few information about the actual structure of the spectra generated by the PSG, one way to test this hypothesis would be to generate the template spectrum without star reflection. If this successfully removes the false positive, this would assert the hypothesis. In order to correct the additional correlated signal, we need to consider the following spectrum for the correlation with the template.

Sstar × C + Sexoplanet Sdetected = (11) Sstar

The division with the non-attenuated spectrum of the host star Sstar will prevent false positives to contribute to the correlation peak. In order to obtain the signal of the non-attenuated star light, one can consider the constructive interference output of the interferometer (see Fig.3). Because Splanet Sstar in this case, the interferometer output is equivalent to Sstar.

Figure 21: Normalized correlations in the case of Fig.20 (orange) and with the correction from (11) (blue).

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The only correlation peak we observe then is the one from the exoplanet, the contribution from the star is corrected.

3.4 The high-pass ﬁlter The information we are interested in are the spectral lines of absorption from the planet. The continuum part and the low-frequency structures then add an unwanted correlated signal. The high-pass ﬁlter we apply in the Fourier space is as follows:

j.ν/ν H(ν) = c (12) 1 + j.ν/νc with ν the spectrum frequency and νc the cutoﬀ frequency.

Figure 22: Example of correlation Figure 23: Example of correlation function without high-pass ﬁltering. function with high-pass ﬁltering with −1 The large spectral structures are over- νc=50µm . The correlation peak ap- whelming. pears.

The ﬁlter enables to isolate the correlation peak from the exoplanet correctly.

3.5 Signal-to-noise ratio of the correlation peak After following the diﬀerent steps to realize the cross-correlation function, we have a result with a correlation peak and a certain noise level (see Fig.23). One should not confuse the photoelectronic noises from sections 2.5/2.7 and the correlation noise we are talking about here. While the ﬁrst ones were depending on physical phenomena, the correlation noise is a mathematical phenomenon from the correlation function. The signal-to-noise ratio corresponds to the ratio between the peak intensity and the root- mean-square of the correlation.

15 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Figure 24: Schema of the SNR calculation protocol.

The root-mean-square is calculated in the ﬁrst and last quarter of the correlation like in [8] (orange squares in Fig.24) in order to avoid the inﬂuence of the central peak. For the example of Fig.24, we have: S SNR = ' 22 (13) exoplanet N

In [8], the authors consider a minimum of SNRexoplanet > 3 and a corresponding velocity that is realistic (∆v ≤ 100km/s) as conditions for a signiﬁcant exoplanet detection.

4 Performance simulation

4.1 Design of the instrument From Fig.7 and the detailed version of the software part in Fig.17, we now have a complete description of the situation. We can make a list of the important parameters corresponding to each part of the situation (object/instrument/software):

Object: • Planetary system (object) :

– Host star radius: R∗ [R ]

– Distance: d∗ [pc]

– Semi-major axis of exoplanet orbit: a⊕ [A.U] Instrument: • Telescopes:

– Baseline: BT [m]

– Eﬀective diameter: DT [m]

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– Optical train temperature: Toptical [K] • Beam combiner: – Null depth: 1/ρ • Spectrometer: – Spectral resolution: R • Photo-detector:

– Bandpass: λ2-λ1 [µm] − – Readout noise variance: Nread [e ] − – Dark current: Idark [e /s]

– Quantum eﬃciency: ηeff

– Number of pixel: npixel – Integration time: t [hours] Software: • High-pass ﬁlter:

−1 – Cutoﬀ frequency: fc [µm ] • Correlation

– Maximum velocity shift: ∆vmax [km/s]

4.2 Conditions of the simulation 4.2.1 Object Planetary system: We simulate a M-star with the same characteristics as Proxima Centauri using the simulations made during a previous project by Mathilde Gardies. Therefore we have:

 R = 0.14R  ∗ d∗ = 1.29pc   a⊕ = 0.032A.U

We assume that for this value of a⊕, the exoplanet is always in the ﬁeld of view of our instrument.

4.2.2 Instrument Telescopes: Among the same four models of space instruments as in [2] already considered in Fig.9, we will consider only one of them as a blueprint for our simulation:

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Figure 25: Four examples of conﬁgurations from nano- to medium-size satellites. Source: [2].

We consider the FKSI-concept in our study since it represents a more ambitious design than the other concepts. This means that we have:

 B = 12.5m  T DT = 0.5m   Toptical = 60K

Beam combiner: The null depth is the ﬁrst variable of our simulation, its value will range from 10−3 to 10−7. Spectrometer: The spectral resolution is the second variable of our simulation, its value will range from 103 to 106. Photo-detector: To estimate the performance of the detector we adopt the state of the art values provided for the NIRSPec instrument on board the JWST Telescope.

Figure 26: NIRSpec NRS1 and NRS2 detector performance summaries. Source: NIRSpec Detector Performance

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If we consider the measured performance of NRS2, we can choose:

 −  Nread ∼ 6e  −  Idark ∼ 0.0058e /s η ∼ 80%  eff  λ1, λ2 = 3, 5µm

The npixel needed in the bandpass [λ1,λ2] increases with the spectral resolution R. We consider that for all values of R, npixel is suﬃcient to sample all the information. For the integration time, a typical value for space missions is t∼ 100hrs.

4.2.3 Software

High-pass filter: The only parameter in our high-pass filter is the cutoff frequency fc. We need to find the optimal values for fc in order to remove effectively the low-frequency structures of the spectra but without damping the absorption lines of higher frequencies.

3 Figure 27: Evolution of the SNR with the cutoﬀ resolution (1/fc) for spectral resolution R=10 (top,left), 104 (top,right), 105 (bottom,left) and 106 (bottom,right). The study is made in the K-band with t=24hrs, 1/ρ=10−6 and the shot noise as the only source of noise.

Fig.27 shows an optimal value for cutoﬀ resolution 1/fc between 0.1 and 1µm. This corresponds −1 to a cutoﬀ frequency fc between 1 and 100µm . This is in agreement with the announced −1 condition from [8]: fc<100µm . −1 We decide to choose arbitrarily fc=50µm for the rest of the study since no clear optimal frequency is emerging between 1 and 100µm−1. Correlation: The full correlation function shows a divergence of the noise level for very high velocity shift.

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Figure 28: Example of a full correlation function.

5 Fig.28 gives an example of how the noise level becomes very unstable for ∆vmax > 10 km/s. Because our spectra are only deﬁned between [1,20]µm, it is normal to have boundary eﬀects for very high velocity shifts. However these instabilities are artefacts and they can alter the noise level and therefore the SNR of the planet. That is the reason why we need to limit the correlation function to a maximum velocity shift ∆vmax in order to avoid instabilities. 4 By choosing a value of ∆vmax ∼5.10 km/s, we limit the correlation function to a stable noise 5 level (cf. Fig.23). This is close to the solution in [8] where ∆vmax ∼3.10 km/s. One might also argue that such velocity speed is unrealistic (>10%.c). However this is just a mathematical value with no physical meaning. Only if the correlation peak was at such speed would the result be questionable.

4.3 Results of the simulation 4.3.1 Without stellar leakage At first, we consider the star light attenuation from the nulling as perfect and without the stellar leakage phenomenon. With the parameters defined in the previous section, we can estimate the impact of every noise on the SNR evolution independently. We can first present the evolution of the SNR when taking each noise into account and then calculate the quadratic difference with the ideal result SNRideal (with no source of noise considered) using the formula:

q 2 2 ∆SNR = SNRideal − SNRnoise (14)

The quadratic diﬀerence will help us ﬁnd which noises have the most impact on the planet detectability.

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Figure 29: Evolution of the SNR Figure 30: Evolution of the quadratic with the spectral resolution for three diﬀerence between the ideal case and sources of photo-electronic noise. We with one source of noise. The ideal consider the star light attenuation case is added as a reference but has C=1/ρ=10−6 (no stellar leakage). no meaning here.

Figure 31: Evolution of the SNR with Figure 32: Evolution of the quadratic the spectral resolution for diﬀerent diﬀerence between the ideal case and background sources. We consider the with one source of noise. The ideal star light attenuation C=1/ρ=10−6 case is added as a reference but has (no stellar leakage). no meaning here.

Without any sources of noise ("ideal" black curves in Fig.29-32), we see that the exoplanet SNR is improving with the spectral resolution. This is a consequence of high-dispersion spectroscopy where increasing spectral resolution enables to reveal more absorption lines from the planet and therefore improves its signal when combining it. In theory we are supposed to have q SNRplanet ∝ log(R). For now, we only have the evolution of the SNR with respect to the spectral resolution R. We also need to investigate the impact of the star light attenuation parameter C on the SNR. For

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this, we will use a 2D representation with the two parameters (C,R) as variables.

Figure 33: Evolution of the SNR in (C,R) without considering any source of noise.

We realize that only the shot noise is C-dependent because it directly infers from the star emission. The other sources such as the background or the instrumental noises are not depending on it.

Figure 35: Evolution of the SNR in Figure 34: Evolution of the SNR in (C,R) considering only the dark cur- (C,R) considering only the shot noise. rent. The result is independent from The stellar leakage is not considered. C.

Background & instrumental behaviour: The fact that the backgrounds and the instrumental noises are independent from C like in Fig.35 is not surprising since their variances are unchanged when diminishing the star emission, therefore their noise levels stay equal. Because the exoplanet signal is not tampered by C either, the SNR is only depending on R. When increasing R, the impact of the instrumental noise increases too for a constant integration time (see Fig.30). This is due to the lower number of photons per pixel when improving the

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spectral resolution, while the instrumental noise levels stay the same. Therefore the noise becomes more important compared to the signal for higher R values. Shot noise behaviour: The reason why the shot√ noise impact increases with C (see Fig.34) is because the shot noise level is proportional to Sstar.C (Section√ 2.3). Increasing C is equivalent to reducing the star attenuation; at some point we have Sstar.C > Sexoplanet and the planet signal becomes overwhelmed by the shot noise from the star. This explains the results from Fig.29-35. We see that the two main sources of noise are the shot noise and the dark current. They are depending on the (C,R) region we are studying: • When C increases: the shot noise becomes dominant • When R increases: the detector noise becomes dominant In these conditions, when taking into account all the sources of noise, we have the remaining exoplanet signal.

Figure 36: Evolution of the SNR is (C,R) when considering all the noises except the stellar leakage.

4.3.2 Comments We observe that in these conditions, the detection of an Earth-like exoplanet is possible even for a star light attenuation of ∼ 10−4. From Fig.8, we see that the initial star/planet contrast between [3,5]µm is around 107. Therefore we successfully relaxed the star light attenuation required to detect an Earth-like exoplanet around Proxima Centauri by a factor of 1000. These results also show the optimal spectral resolution (or the spectral resolution with the maximum SNR) for an observation with our parameters. In our case, R ∼ 10.000 seems to represent the optimal resolution. However one should also consider the increasing cost and technical diﬃculty for high R values.

4.3.3 With stellar leakage In the previous section, we considered every limitations for the planet signal independently. However we can’t consider the eﬀects of the star leakage alone because it only caps the maximum

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attenuation reachable. From Fig.33, we saw that the ideal SNR of the planet is C-independent which makes it impossible to evaluate the impact of the stellar leakage. A solution would be to first consider the shot noise and then apply the stellar leakage limitation, since the shot noise is C-dependent. The quantum efficiency is another limitation like the stellar leakage whose impact can’t be estimated alone. We can use the same technique with the shot noise to observe its influence. However, because the difference is minimal compared to the rest, we show the results in Ap- pendix 2. For two telescopes separated by a baseline of B=12.5m, the star light attenuation is limited to C'2,7.10−4 for Proxima Centauri using (7).

Figure 37: Evolution of the SNR in Figure 38: Evolution of the SNR in (C,R) considering only the shot noise. (C,R) considering the shot noise and the stellar leakage.

The two results from Fig.37-38 show that the shot noise becomes the main source of noise when considering the stellar leakage. Indeed if the star light attenuation is capped to ∼ 10−4, then the shot noise will remain dominant for any (C,R) point we are considering. With the parameters from section 4.2, the detection of an Earth-like exoplanet around Proxima Centauri is possible but the stellar leakage is limitating a lot its intensity.

4.4 Conclusion The detection of Earth-like exoplanets with a small angular separation and a high contrast is then possible considering the method of high-dispersion nulling interferometry. However phenomena like stellar leakage and detector noise are drastically deteriorating the planet detectability. Therefore, to improve the performances of our instrument we have two options: increasing the number of telescopes to reduce the stellar leakage [13] and consider a detector with very low readout noise and dark current. A new generation of detectors, the MKIDs, are showing promising results in this ﬁeld [14].

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5 Study of a photonic device for the nulling

5.1 Introduction IPAG is currently exploring diﬀerent concepts to demonstrate experimentally high dispersion nulling interferometry. This instrumental research relies on laboratory prototyping and later precursor instruments that will be used for actual observations. One of the speciﬁc research avenues followed is the use of photonic components to carry some of the most critical functions of a nulling interferometer namely the beam combination function. Among the high diversity of photonic technologies those based on LiNbO3 crystals represent a particularly interesting avenue since they are 1) more transmissive in the [2-5]µm bandpass, which is largely unexplored in astronomy, 2) they allow the integration of active phase controlling functions within the photonic circuit. This latter point is particularly critical in the nulling combiner since one has to maintain the instrument integrating at the destructive interference point. The goal for this part of my project was to explore the quality of the interferometric signal that could be obtained with a LiNbO3 two telescopes circuit in order to explore the potential of the technology for nulling interferometry.

5.2 Component description As stated in the introduction the component is a two telescopes circuit which means that it includes two input waveguides. With only two telescopes, several architectures can already be imagined to realize the beam combiner depending on the interferometric fringe encoding scheme. We will study one corresponding to a setup of two inputs and four outputs. The component is called 2T ABCD and uses a double Mach-Zehnder interferometer waveguide.

Figure 39: Schema of the 2T ABCD photonic device with two sets of electrodes. Source: [4]

The two inputs (E1,E2) give four outputs (S1-S4) called ABCD with a phase diﬀerence of π/2 between two consecutive ones (see Fig.40).

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Figure 40: Schema of the phase diﬀerence between the four outputs of the device. Source: [5].

The circuit also includes two sets of electrodes (DC1,DC2) and (DC3-DC6) which allows a phase modulation when applying a voltage to them. The main electrodes are the (DC1,DC2): when applying a voltage V12 between them, we generate a phase diﬀerence φ12 corresponding to a change in refractive index by the Pockels eﬀect:

2π 1 3 V12 φ12 = − n1.r33. .Lelec (15) λ0 2 delec with λ0 the electromagnetic wavelength of the light in m, n1 the passive refractive index in the direction of propagation, r33 the highest electro-optic coeﬃcient of the crystal, delec the electrodes separation and Lelec the electrode length. This phase diﬀerence can compensate the instabilities from the optical delay lines and stabilize the instrument at the destructive interference point. In addition with its electro-optical properties, the LiNbO3 also enables the building of single mode guides around 1.55µm wavelength with a single transmitted polarization [4].

5.3 Experimental setup The component is characterized on the optical bench SILVI which simulates a source at inﬁnity being sampled by a network of telescopes (up to four telescopes in H-band and K-band).

Figure 41: Schema of the SILVI experimental setup. Source: [6]

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We have two types of source at our disposal, a laser emitting at λ=1550.54nm and a white source emitting at large spectral band called ASE (Ampliﬁed Spontaneous Emission) or SLED (Superluminescent Diode).

Figure 42: ASE spectrum from the constructor.

We place the photonic component just after the recombiner. Then the process for the optical bench is as follow: 1. Using an optical fiber, we introduce the object as a point source Laser (monochromatic source) or ASE/SLED (polychromatic source), 2. The parabolic mirror creates the collimated beam sent to the telescopes, 3. The different telescopes gather a part of the light into their corresponding fibers. Each telescope can be moved to change the optical delay line and then scan for the optimal path length. 4. The fibers are gathered in a v-groove and can be aligned with the photonic circuit, 5. The spectro-imager is designed with 2 sets of lenses and a grating to diffract the light and focus it on the detector,

5.4 Interferogram in the monochromatic case In the monochromatic case (with the Laser source) we have a well known formula to describe the interferogram: q Itot = I1 + I2 + 2. I1.I2V cos(φ) + Bg (16) with I1 and I2 being the intensities in inputs 1 and 2, V the contrast, Bg the background ﬂux and φ the relative phase given by:

φ = φ0 + 2πσ(x − x0) + φ12 (17) with Φ0 the initial phase (see Fig.40), σ the wave number, x the Optical Path Delay (OPD), x0 the initial OPD and φ12 the phase diﬀerence from (15). The contrast V of the interferogram is the parameter which gives us the level of nulling we can achieve. In order to obtain the form of the interferogram and its contrast, we scan the relative phase φ either with external modulation or internal modulation.

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With external modulation: An OPD x is generated via a relative translation of the telescopes along the optical axis from one another δ. This OPD is equal to

x = n.δ (18) with n the refractive index of the environment (air in our case). This will change the relative phase according to (17).

Figure 43: Interferograms for the four outputs in the Laser case with external modulation.

With internal modulation: A phase difference is generated when applying a voltage V12 between the electrodes (DC1,DC2). This induces a relative phase difference φ12 according to (15) by the Pockels effect.

This phase variation enables to scan the interferogram for each four outputs. Fig.43 is an example of scan with external modulation. The results from Fig.43 conﬁrm that our four outputs ABCD have their relative phase diﬀer- ences in agreement with Fig.40. The pixels 1,2,3 and 4 corresponds to the output B,D,A and C respectively.

5.5 The contrast To obtain the contrast from these interferometers, we can use a corrected version with an intensity corresponding to

Itot − I1 − I2 − Bg Ic = √ + 1 = 1 + V cos(φ) (19) 2 I1.I2 This way, the contrast V of the interferogram is given by: I − I V = c,max c,min (20) Ic,max + Ic,min

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Figure 44: Corrected interferograms for outputs in Fig.43 using (19).

Using the corrected interferograms of Fig.44 and (20), we obtain the contrast values for each of them: VB ' 0.896; V2 ' 0.864; V3 ' 0.898; V4 ' 0.881 (21) In practice, we consider that the outputs ABCD are four points of the same corrected interferogram (see Fig.40). This way, we can obtain the contrast of this interferogram with just one measure of the ABCD outputs thanks to their relative phase diﬀerences. For a contrast V ∼ 0.9, it implies a null depth of N ∼ 0.1. This value of null depth is far from the levels of star attenuation required for an Earth-like detection (see Section 4.3). The main sources of our null depth limitation are: • The relative polarization, • The chromatism, • The photometric equilibrium, • The relative phase stability. Because our waveguides only allow one polarization to be transmitted, we can ignore the relative polarization limitation.

5.5.1 The chromatism

For a laser source, their is no limitation from chromatism since the band width ∆λ λ0, this means that the coherency envelope Λ of the interferogram is very large:

λ2 Λ ≡ 0 (22) ∆λ However in the case of the white source, Λ becomes far smaller because of the large band ∆λ. −1 This is due to the fact that the relative phase φ ∝ λ0 , so the period of the interferogram is diﬀerent for every wavelength. This causes a loss of coherency and also a loss of amplitude in the interferences.

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Figure 45: Example for a polychromatic source. It shows the individual monochromatic re- sponses (left) and the resulting fringe pattern for a polychromatic source (right). Source:[1].

This result is only possible if there is no dispersion diﬀerence or length diﬀerence in the system. Then the central fringe is achromatic (see Fig.45).

Figure 46: Observed interferogram for the SLED source. We can observe the impact of the coherency envelope.

In Fig.45-46, we see that the interferometric contrast is decreasing due to the loss of temporal coherence. This means that the device needs to always be centered on the achromatic central fringe in order to have the highest contrast achievable. Before considering the chromatism limit for the null depth, one should study the Laser case ﬁrst in order to only consider the photometric and phase limitations. Only after dealing with this should one consider the chromatism eﬀect.

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5.5.2 The photometric equilibrium

If we consider a fractionnal intensity mismatch ∆ between I1 and I2, for example like:

I2 = I1(1 + ∆) (23)

Then using (16), we have the minimum and maximum values of the interferogram:

( √ Imin = 2 + ∆ − 2√1 + ∆ Imax = 2 + ∆ + 2 1 + ∆

Which means we can express the null depth 1/ρ depending on ∆: √ I 2 + ∆ − 2 1 + ∆ ∆2 N = min = √ ' (24) Imax 2 + ∆ + 2 1 + ∆ ∆1 16 Then if we want our component to realize a certain null depth, we have to be sure that the two inputs have a limited mismatch. For example, in order to have a null depth of N = 10−4, this requires ∆ ≤ 4%. In order to ensure such precision, we can use intensity modulators on our input ﬁbers. This way, it becomes possible to control the input intensities and their mismatch.

Figure 47: Evolution of the null depth with the intensity mismatch between the two inputs.

5.5.3 The relative phase instability The experiment is using optical fibers to guide the light and conserve its properties effectively. However these optical fibers are also quite sensitive to their environment like temperature or vibrations. This causes the relative path difference between the two inputs to suffer instabilities over time, which affects the initial phase φ0.

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Figure 48: Evolution of the detected signal from the four outputs of the component over time without any modulation.

Fig.48 demonstrates that even without a phase modulation, the optical ﬁbers sensitivity gener- ates a phase instability with a characteristic time of several seconds. A phase control using the electro-optic properties of LiNbO3 would have to monitor and correct these phase drifts and ensure a relative phase stability.

5.6 Comments The interferograms we obtain in the Laser case and the white source case successfully demonstrated the ability of our experiment to realise nulling interferometry. However the levels of null depth we obtain in the Laser case are far from the requirements for stellar attenuation from Section 4.3. To improve the performances of our experiment, a future project should first focus on the photometric equilibrium and phase instabilities to obtain better null depth in the Laser case. For the photometric equilibrium, an intensity modulator has been placed before one of the inputs to reduce the intensity mismatch. However no significant results were found, mainly because of the absence of a real-time reading of the output signal which prevents from creating a feedback loop to control the intensity mismatch. For the phase instabilities, the use of OPD sensors in the optical fibers can give a real-time control of phase drifts. The active phase controlling functions, using the electrodes of the circuit, can then correct this drift effectively.

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6 Conclusion

In the first part of the project, the study for the combination of high-dispersion spectroscopy and nulling interferometry shows positive results in terms of Earth-like detectability. We obtain a relaxation for the star light attenuation requirement by a factor of 1000. Thanks to this relaxation, the nulling interferometry can now achieve reasonable levels of stellar attenuation (∼ 10−4) which allows the detection of an Earth-like exoplanet around Proxima Centauri. How- ever in order to increase the planet signal and search for rocky exoplanets in distant planetary systems, one needs to limit the impact of the two major sources of noise. The first one is the shot noise that mainly comes from stellar leakage. This can be limited by combining more than two telescopes to increase the angular depth of the null [13]. The second one is the detector noise. A new generation of detectors, the MKIDS, are expected to reach a very low level of dark current and readout noise which could be a technical solution if this is confirmed [14]. These detectors are also expected to improve the spectral resolution that can be achieved, reducing the cost and technical difficulties for high-spectral resolution.

In the second part of the project, the study of the photonic component with a Laser source showed its ability to generate the expected interferogram with its corresponding contrast. How- ever, the contrast value we found V ∼ 0.9 gives a null depth performance for nulling interferometry of N ∼ 0.1. According to the first part of our study, this is not sufficient for Earth-like detection: the objective is a null depth of N ∼ 10−4. To achieve such goal, several constraints need to be placed on the precision and stability of our experiment and specifically for the intensity mismatch in input and the phase drifts. Two features can thus be added in our experiment in order to improve the nulling performance. The first ones are intensity modulators for the two component inputs in order to reduce the intensity mismatch. However this solution requires a real-time reading of the output signal to find the minimal mismatch position. The other features are Optical Path Delay sensors to detect and control the phase drifts from the fibers sensitivity to their environment. The correction of these phase drifts is done by using the electro-optical properties of LiNbO3. Two sets of electrodes are placed in the circuit and allow a phase modulation with an electric signal.

If properly placed and controlled, these features should be able to prove if the photonic device can achieve the required null depth for Earth-like observation. If this is the case, this device could be a part of a future precursor project at IPAG to demonstrate experimentally the performance of high-dispersion nullin interferometry.

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References

[1] Chris Haniff. An introduction to the theory of interferometry. NAR, 51(8-9):565–575, October 2007. [2] Colin Dandumont, Denis Defrère, Jens Kammerer, Olivier Absil, Sascha P. Quanz, and Jérôme Loicq. Exoplanet detection yield of a space-based Bracewell interferometer from small to medium satellites. Journal of Astronomical Telescopes, Instruments, and Systems, 6:035004, July 2020. [3] Pavel Gabor. A study of the performance of a nulling interferometer testbed preparatory to the Darwin mission. PhD thesis, Institut d’Astrophysique Spatiale, Universite Paris XI, September 2009. [4] S. Heidmann, O. Caballero, A. Nolot, M. Gineys, T. Moulin, A. Delboulbé, L. Jocou, J. B. Le Bouquin, J. P. Berger, and G. Martin. Two telescopes ABCD electro-optic beam combiner based on lithium niobate for near infrared stellar interferometry. In Mario Bertolotti, editor, Nonlinear Optics and Applications V, volume 8071 of Society of Photo-Optical In- strumentation Engineers (SPIE) Conference Series, page 807108, June 2011. [5] Samuel Heidmann. Composants actifs en optique intégrée pour l’interférométrie stellaire dans le moyen infrarouge. Theses, Université de Grenoble, December 2013. [6] Laurent Jocou. Conception, intégration et caractérisation d’un simulateur inter- férométrique pour l’astronomie. Mémoire, CNAM, 2006-2007. [7] E. Serabyn, B. Mennesson, S. Martin, K. Liewer, and J. Kühn. Nulling at short wavelengths: theoretical performance constraints and a demonstration of faint companion detection inside the diffraction limit with a rotating-baseline interferometer. MNRAS, 489(1):1291–1303, October 2019. [8] Ji Wang, Dimitri Mawet, Garreth Ruane, Renyu Hu, and Björn Benneke. Observing Exoplanets with High Dispersion Coronagraphy. I. The Scientific Potential of Current and Next-generation Large Ground and Space Telescopes. AJ, 153(4):183, April 2017. [9] G. L. Villanueva, M. D. Smith, S. Protopapa, S. Faggi, and A. M. Mandell. Planetary Spectrum Generator: An accurate online radiative transfer suite for atmospheres, comets, small bodies and exoplanets. JQSRT, 217:86–104, September 2018. [10] T. Kelsall, J. L. Weiland, B. A. Franz, W. T. Reach, R. G. Arendt, E. Dwek, H. T. Freuden- reich, M. G. Hauser, S. H. Moseley, N. P. Odegard, R. F. Silverberg, and E. L. Wright. The COBE Diffuse Infrared Background Experiment Search for the Cosmic Infrared Back- ground. II. Model of the Interplanetary Dust Cloud. APJ, 508(1):44–73, November 1998. [11] Alain Léger, Denis Defrère, Fabien Malbet, Lucas Labadie, and Olivier Absil. Impact of ηEarth on the Capabilities of Affordable Space Missions to Detect Biosignatures on Extrasolar Planets. APJ, 808(2):194, August 2015. [12] Eugene Serabyn. Nulling interferometry: symmetry requirements and experimental results. In Pierre Léna and Andreas Quirrenbach, editors, Interferometry in Optical Astronomy, volume 4006 of Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, pages 328–339, July 2000.

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[13] Olivier Guyon, Bertrand Mennesson, Eugene Serabyn, and Stefan Martin. Optimal Beam Combiner Design for Nulling Interferometers. PASP, 125(930):951, August 2013. [14] Benjamin Mazin, Jeb Bailey, Jo Bartlett, Clint Bockstiegel, Bruce Bumble, Gregoire Coif- fard, Thayne Currie, Miguel Daal, Kristina Davis, Rupert Dodkins, Neelay Fruitwala, Ne- manja Jovanovic, Isabel Lipartito, Julien Lozi, Jared Males, Dimitri Mawet, Seth Meeker, Kieran O’Brien, Michael Rich, Jenny Smith, Sarah Steiger, Noah Swimmer, Alex Walter, Nick Zobrist, and Jonas Zmuidzinas. MKIDs in the 2020s. In Bulletin of the American Astronomical Society, volume 51, page 17, September 2019.

35 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Appendix 1

Thermal emission from the optical train According to [2], the optical train acts as a blackbody with an emissivity around 25%. We can use the Planck’s law of blackbody radiation to compute the number of photons reaching the detector from the spectral radiance density B(ν,T):

2hν3 1 B(ν, T ) = 2 hν (25) c e kB T − 1 with B the spectral radiance density in W/str/m2/Hz, T the black body temperature in K, h the Planck’s constant in J.s, c the light speed in vacuum in m/s, kB the Boltzmann’s constant in J/K and ν the electromagnetic frequency in Hz. We consider the emitting area A to be equivalent to the collecting area of the instrument. If we consider the integration time t in s and the solid angle of the detector Ω in str, the number of photons between two frequencies ν1 and ν2 with ν1 < ν2 is:

Z ν2 B(ν, T ) Nph(T, ν1, ν2) = t.A. Ω(ν).dν (26) ν1 hν The solid angle of the detector is given by the diﬀraction of the light through the instrument with diameter D: 1, 22.λ!2 Ω(λ) = π. sin2(θ) = π. (27) D with sin(θ) the numerical aperture of the detector. Using (26), we can simulate the spectrum of the thermal emission for three temperatures:

Figure 49: Example for a setup of two telescopes with diameter 1m, a resolution of R=103, an integration time of t=100hrs and an optical train temperature of Toptical=60, 100 and 150K

36 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Appendix 2

Impact of the quantum eﬃciency

The quantum efficiency (QE) we consider from Fig.26 is estimated at around ηeff ∼0.8. Because we can’t evaluate the impact of it independently from other sources of noise, we have to consider first the impact of the shot noise. The difference of influence between the shot noise with and without the quantum efficiency values its effect.

Figure 50: Evolution of the SNR with the spectral resolution considering the shot noise with and without the quantum eﬃciency. The star light attenuation is at C=1/ρ=10−6.

By modifying the formula (14) into:

q 2 2 ∆SNRQE = SNRWithout QE − SNRWith QE We obtain the quadratic diﬀerence corresponding to the quantum eﬃciency. This way we can evaluate its impact on the planet signal.

Figure 51: Evolution of the quadratic diﬀerence between the ideal case and with the shot noise (blue) and between the shot noise without and with quantum eﬃciency (red). The ideal case is added as a reference but has no meaning

This result shows that the quantum eﬃciency is equivalent to the zodiacal noise in terms of SNR limitation. We can therefore ignore it as a main source of noise.

37 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Appendix 3

Gantt Diagram

38 Detecting Earth-like exoplanets using high-dispersion nulling interferometry Garreau Germain

Abstract

The detection of Earth-like exoplanets and the characterization of their atmospheres is a challenge one needs to solve to assess their habitability and the presence of life in the universe. If this challenge is still unresolved today, even in the era of giant telescopes, it is mainly because of the very high contrast between these exoplanets and their host star and also their proximity. To overcome both of these constraints, a new method combining high-dispersion spectroscopy and nulling interferometry has been imagined. The idea is to use the nulling interferometry to attenuate the star light emission and detect the inner rocky planets with a high angular resolution. The high-dispersion spectroscopy is increasing the exoplanet detectability signiﬁ- cantly which enables to relax the star attenuation requirement for an Earth-like observation. Our simulation made for an exoplanet similar to the Earth orbiting Proxima Centauri is giving a condition for the star attenuation ∼ 10−4 to detect it. Given this condition, we are able to evaluate the unability of a photonic device at our disposal to achieve such performance without dealing with its limitations. If a future project manage to overcome these limitations, this device could be part of a precursor instrument at IPAG to demonstrate experimentally the performance of high-dispersion nulling interferometry.

La détection de planètes de type Terre et la caractérisation de leurs atmosphères est un défi qu’il est nécessaire de résoudre afin de pouvoir estimer leur habitabilité et la présence de vie dans l’Univers. Si un tel défi n’a toujours pas pu être résolu à ce jour, même à l’ère des télescopes géants, c’est à la fois à cause du très haut contraste entre ces exoplanètes et leur étoile et également leur grande proximité. Pour surmonter ces deux contraintes, une nouvelle méthode combinant la haute dispersion spectrale et l’interférométrie annulante a été imaginée. Le principe est d’utiliser l’interférométrie annulante afin d’atténuer le signal de l’étoile tout en ayant la résolution angulaire suffisante pour détecter les planètes rocheuses proches. La haute dispersion spectrale permet d’augmenter significativement la détectabilité de l’exoplanète, ce qui assouplit l’exigence en terme d’atténuation de l’étoile pour observer une planète de type Terre. Nous considérons pour notre simulation une exoplanète similaire à la Terre orbitant Proxima Centauri, cela nous donne une attenuation de l’étoile minimale de ∼ 10−4 pour détecter la planète. Cette condition nous permet de réaliser l’incapacité d’un composant photonique en notre possession d’atteindre cette performance sans considérer ses facteurs limitants. Si un futur projet parvient à surmonter ces limitations, ce composant pourrait faire partie d’un intrument précurseur à l’IPAG visant à démontrer expérimentalement les performances de la haute dispersion spectrale couplée à l’interférométrie annulante.